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Numerical simulation

Numerical simulations presented in this work were performed on a single-processor workstation (2.5GHz, 1Gb RAM). We performed a direct numerical simulation, integrating the dynamical equations of motion (4) and (5) using pseudo-spectral method with resolution of $ 256\times256$ wavenumbers. Numerical integrator used for advancing in time was RK7(8) presented in [31]. Time step was $ \frac{T_{min}}{35}$ where $ T_{min}$ is the period of the shortest wave on the axis. Approximate processor time for this work was 4.5 weeks.

In our numerical experiment, we force the system in the $ k$-space ring $ k_* < k < k^*$ with $ k_* =6$ and $ k^* = 9$. This ring is located at the low wavenumber part of the $ k$-space in order to generate energy cascade toward large $ k$'s, but we deliberately avoid forcing even longer waves ($ k<6$) because our experience shows that this would lead to undesirable strong anisotropic effects. In the ring, we fix the shape to coincide with the ZF spectrum, $ <\vert a_{\mathbf{k}}(t)\vert^{2}>\sim
k^{-4}$, and hence we set $ \vert\eta _{\mathbf{k} }\vert \sim k^{-7/4}$, $ \vert\Psi _{\mathbf{k}}\vert \sim k^{-9/4}$. These fixed amplitudes were then multiplied by random phase factors. Thus, surface $ \eta
_{\mathbf{k}}$ and velocity potential $ \Psi_{\mathbf{k}}$ were set to $ 2\pi ^{3} \, e^{i \theta _{\mathbf{k}}} \ast k^{\alpha
_{\mathbf{k}}}$, where $ \theta _{\mathbf{k}}$ were uniformly distributed in $ [0,2\pi
]$ and

$\displaystyle \alpha _{k}=x\left\{ \begin{array}{cc} \left[ 1+\left( \frac{k_{\...
.../4} & \hbox{if} \;k\in \left( k^{\ast },\frac{N}{2}\right) \end{array} \right\}$    

where $ x=-7/4$ for the surface and $ x=-9/4$ for velocity potential. Damping was applied in both small and large wavenumber regions. At small wavenumbers inside the forcing ring, we applied an adaptive damping to prevent formation of undesirable ``condensate'' which could spoil isotropy and locality of scale interactions. At large wavenumbers, damping is needed to absorb the energy cascade and, therefore, to avoid ``bottleneck'' spectrum accumulation near the cutoff wavenumber. In our simulations, we implemented the damping as a low-pass filter $ \gamma _{\mathbf{k}}$ applied to the $ k$-space variables at each time step. The damping function had the form

$\displaystyle \gamma _{\mathbf{k}}=\left\{ \begin{array}{cc} 5\left( k-6\right)...
... \left[ 6,64\right] \\ 0.028\left( k-64\right) ^{2} & k>64 \end{array} \right\}$    

Nonlinearity parameter was set to $ \varepsilon =2\cdot 10^{-2}$, which is a sufficient value to produce a resonance broadening for supporting energy cascade.

next up previous
Next: Results Up: Discreteness and its effect Previous: Resonant interaction in discrete
Dr Yuri V Lvov 2007-01-16